Convolutional dictionary learning (CDL) aims to learn a structured and shift-invariant dictionary to decompose signals into sparse representations. While yielding superior results compared to traditional sparse coding methods on various signal and image processing tasks, most CDL methods have difficulties handling large data, because they have to process all images in the dataset in a single pass. Therefore, recent research has focused on online CDL (OCDL) which updates the dictionary with sequentially incoming signals. In this article, a novel OCDL algorithm is proposed based on a local, slice-based representation of sparse codes. Such representation has been found useful in batch CDL problems, where the convolutional sparse coding and dictionary learning problem could be handled in a local way similar to traditional sparse coding problems, but it has never been explored under online scenarios before. We show, in this article, that the proposed algorithm is a natural extension of the traditional patch-based online dictionary learning algorithm, and the dictionary is updated in a similar memory efficient way too. On the other hand, it can be viewed as an improvement of existing second-order OCDL algorithms. Theoretical analysis shows that our algorithm converges and has lower time complexity than existing counterpart that yields exactly the same output. Extensive experiments are performed on various benchmarking datasets, which show that our algorithm outperforms state-of-the-art batch and OCDL algorithms in terms of reconstruction objectives.
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